To appear in Grazer Philosophische Studien 2007 and in Christian Beyer and Alex Burri

(eds.), Philosophical Knowledge — Its Possibility and Scope, Amsterdam: Rodopi.

Philosophical Knowledge and Knowledge of Counterfactuals*

Timothy Williamson

ABSTRACT

Metaphysical modalities are definable from counterfactual conditionals, and the

epistemology of the former is a special case of the epistemology of the latter. In

particular, the role of conceivability and inconceivability in assessing claims of

possibility and impossibility can be explained as a special case of the pervasive role of

the imagination in assessing counterfactual conditionals, an account of which is sketched.

Thus scepticism about metaphysical modality entails a more far-reaching scepticism

about counterfactuals. The account is used to question the significance of the distinction

between a priori and a posteriori knowledge.

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§0. Philosophers characteristically ask not just whether things are some way but whether

they could have been otherwise. What could have been otherwise is metaphysically

contingent; what could not is metaphysically necessary. We have some knowledge of

such matters. We know that Henry VIII could have had more than six wives, but that

three plus three could not have been more than six. So there should be an epistemology of

metaphysical modality.

The differences between metaphysical necessity, contingency and impossibility

are not mind-dependent, in any useful sense of that tantalizing phrase. Thus they are not

differences in actual or potential psychological, social, linguistic or even epistemic status

(Kripke 1980 makes the crucial distinctions). One shortcut to this conclusion uses the

plausible idea that mathematical truth is mind-independent. Since mathematics is not

contingent, the difference between truth and falsity in mathematics is also the difference

between necessity and impossibility; consequently, the difference between necessity and

impossibility is mind-independent. The difference between contingency and non-

contingency is equally mind-independent; for if C is a mind-independently true or false

mathematical conjecture, then one of C and its negation conjoined with the proposition

that Henry VIII had six wives forms a contingently true conjunction while the other

forms an impossible conjunction, but which is which is mind-independent. To emphasize

the point, think of the mind-independently truth-valued conjecture as evidence-

transcendent, absolutely undecidable, neither provable nor refutable by any means. Thus

the epistemology of metaphysical modality is one of mind-independent truths.

Nevertheless, doubts begin to arise. Although philosophers attribute metaphysical

necessity to mathematical theorems, what matters mathematically is just their truth, not

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their metaphysical necessity: mathematics does not need the concept of metaphysical

necessity. Does metaphysical modality really matter outside philosophy? Even if

physicists care about the physical necessity of the laws they conjecture, does it matter to

physics whether physically necessary laws are also metaphysically necessary? In ordinary

life, we care whether someone could have done otherwise, or whether disaster could have

been averted, but the kind of possibility at issue there is far more narrowly circumscribed

than metaphysical possibility, by not prescinding from metaphysically contingent initial

conditions. He could not have done otherwise because he was in chains, even though it

was metaphysically contingent that he was in chains. Does ‘could have been’ ever

express metaphysical possibility when used non-philosophically?

If thought about metaphysical modality is the exclusive preserve of philosophers,

so is knowledge of metaphysical modality. The epistemology of metaphysical modality

tends to be treated as an isolated case. For instance, much of the discussion concerns how

far, if at all, conceivability is a guide to possibility, and inconceivability to impossibility

(Gendler and Hawthorne 2002 has a sample of recent contributions to this debate). The

impression is that, outside philosophy, the primary cognitive role of conceiving is

propaedeutic. Conceiving a hypothesis is getting it onto the table, putting it up for serious

consideration as a candidate for truth. The inconceivable never even gets that far.

Conceivability is certainly no good evidence for the restricted kinds of possibility that we

care about in natural science or ordinary life. We easily conceive particles violating what

are in fact physical laws, or the man without his chains. On this view, conceiving, outside

philosophy, is not a faculty for distinguishing between truth and falsity in some domain,

but rather a preliminary to any such faculty. Although there are truths and falsehoods

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about conceivability and inconceivability, they concern our mental capacities, whereas

metaphysical modalities are supposed to be mind-independent. They are not contingent

on mental capacities, because not contingent on anything (at least if we accept the

principles of the modal logic S5, that the necessary is necessarily necessary and the

possible necessarily possible). When philosophers present conceiving as a faculty for

distinguishing between truth and falsity in the domain of metaphysical modality, that

looks suspiciously like some sort of illicit projection or unacknowledged fiction: at best,

attributions of metaphysical modality would lack the cognitive status traditionally

ascribed to them (compare Blackburn 1987; Craig 1985; Wright 1989). The apparent

cognitive isolation of metaphysically modal thought makes such suspicions hard to allay.

Presenting it as sui generis suggests that it can be surgically removed from our

conceptual scheme without collateral damage. If it can, what good does it do us? In

general, the postulation by philosophers of a special cognitive capacity exclusive to

philosophical or quasi-philosophical thinking looks like a scam.

Humans evolved under no pressure to do philosophy. Presumably, survival and

reproduction in the stone age depended little on philosophical prowess (dialectical skill

was probably no more effective then as a seduction technique than it is now). Any

cognitive capacity that we have for philosophy is a more or less accidental byproduct of

other developments. Nor are psychological dispositions that are non-cognitive outside

philosophy likely suddenly to become cognitive within it. We should expect the cognitive

capacities used in philosophy to be cases of general cognitive capacities used in ordinary

life, perhaps trained, developed and systematically applied in various special ways, just

as the cognitive capacities that we use in mathematics and natural science are rooted in

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more primitive cognitive capacities to perceive, count, reason, discuss …. In particular, a

plausible non-sceptical epistemology of metaphysical modality should subsume our

capacity to discriminate metaphysical possibilities from metaphysical impossibilities

under more general cognitive capacities used in ordinary life.

I will argue that the ordinary cognitive capacity to handle counterfactual

conditionals carries with it the cognitive capacity to handle metaphysical modality. §1

illustrates with examples our cognitive use of counterfactual conditionals. §2 sketches the

beginnings of an epistemology of such conditionals. §3 explains how they subsume

metaphysical modality. §4 discusses some objections. §5 briefly raises the relation

between metaphysical possibility and the restricted kinds of possibility that seem more

relevant to ordinary life. Philosophers’ ascriptions of metaphysical modality are far more

deeply rooted in our ordinary cognitive practices than most sceptics realize.

§1. We start with a well-known example that proves the term ‘counterfactual conditional’

misleading. As Alan Ross Anderson pointed out (1951: 37), a doctor might say:

(1) If Jones had taken arsenic, he would have shown just exactly those symptoms

which he does in fact show.

Clearly, (1) can provide abductive evidence by inference to the best explanation for its

antecedent (see Edgington 2003: 23-7 for more discussion):

(2) Jones took arsenic.

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If further tests subsequently verify (2), they confirm the doctor’s statement rather than in

any way falsifying it or making it inappropriate. If we still call subjunctive conditionals

like (1) ‘counterfactuals’, the reason is not that they imply or presuppose the falsity of

their antecedents.

Of course, what (2) explains is not the trivial necessary truth that Jones shows

whatever symptoms he shows. What is contingent is that Jones shows exactly those

symptoms which he does in fact show — he could have shown other symptoms, or none

— and, given (1), (2) explains that contingent truth.

While (1) provides valuable empirical evidence, the corresponding indicative

conditional does not (Stalnaker 1999: 71):

(1I) If Jones took arsenic, he shows just exactly those symptoms which he does in fact

show.

We can safely assent to (1I) without knowing what symptoms Jones shows, since it holds

whatever they are. Informally, (1) is non-trivial because it depends on a comparison

between independently specified terms, the symptoms which Jones would have shown if

he had taken arsenic and the symptoms which he does in fact show; by contrast, (1I) is

trivial because it involves only a comparison of his symptoms with themselves. Thus the

process of evaluating the ‘counterfactual’ conditional requires something like two files,

one for the actual situation, the other for the counterfactual situation, even if these

situations turn out to coincide. No such cross-comparison of files is needed to evaluate

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the indicative conditional. Of course, when one evaluates an indicative conditional while

disbelieving its antecedent, one must not confuse one’s file of beliefs with one’s file of

judgments on the supposition of the antecedent, but that does not mean that cross-

referencing from the latter file to the former can play the role that it did in the

counterfactual case.

Since (1) constitutes empirical evidence, its truth was not guaranteed in advance.

If Jones had looked suitably different, the doctor would have had to assert the opposite

counterfactual conditional:

(3) If Jones had taken arsenic, he would not have shown just exactly those symptoms

which he does in fact show.

From (3) we can deduce the falsity of its antecedent. For modus ponens is generally

agreed to be valid for counterfactual conditionals. Thus (2) and (3) yield:

(4) Jones does not show just exactly those symptoms which he does in fact show.

Since (4) is obviously false, we can deny (2) given (3).

The indicative conditional corresponding to (3) is:

(3I) If Jones took arsenic, he does not show just exactly those symptoms which he

does in fact show.

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To assert (3I) would be like saying ‘If Jones took arsenic, pigs can fly’. Although a very

confident doctor might assert (3I), on the grounds that Jones certainly did not take

arsenic, that certainty may in turn be based on confidence in (3), and therefore on the

comparison of actual and counterfactual situations.

Could a Bayesian account dispense with the counterfactual conditionals in favour

of conditional probabilities? Consider the simple case in which we completely trust the

doctor who asserts (1). Before the doctor speaks, we are certain what symptoms Jones

shows but agnostic over the characteristic symptoms of arsenic poisoning. We want to

update our probability for his having taken arsenic on evidence from the doctor, in

Bayesian terms by conditionalizing on it. The doctor cannot simply tell us what

probability to assign, because we may have further relevant evidence unavailable to the

doctor, for example about Jones’s character. We need the doctor to say something which

we can use as evidence; (1) exactly fits the bill (of course, our evidence also includes the

fact that the doctor asserted (1), but in the circumstances we can treat (1) itself as the

relevant part of our evidence). It may even do better than a non-modal generalization

such as ‘Jones showed exactly those symptoms which everyone who takes arsenic

shows’: for the symptoms may vary with bodily characteristics of the victim, and through

long experience the doctor may be able to judge what symptoms Jones would have shown

if he had taken arsenic without being able to articulate a suitable generalization. Any

Bayesian account depends on an adequately varied stock of propositions to act as bearers

of probability, as evidence or hypotheses. Sometimes that range has to include

counterfactual conditionals.

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We also use the notional distinction between actual and counterfactual situations

to make evaluative comparisons:

(5) If Jones had not taken arsenic, he would have been in better shape than he now is.

Such counterfactual reflections facilitate learning from experience; one may decide never

to take arsenic oneself. Formulating counterfactuals about past experience is empirically

correlated with improved future performance in various tasks.1

Evidently, counterfactual conditionals give clues to causal connections. This point

does not commit one to the ambitious programme of analysing causality in terms of

counterfactual conditionals (Lewis 1973b, Collins, Hall and Paul 2004), or counterfactual

conditionals in terms of causality (Jackson 1977). If the former programme succeeds, all

causal thinking is counterfactual thinking; if the latter succeeds, all counterfactual

thinking is causal thinking. Either way, the overlap is so large that we cannot have one

without much of the other. It may well be over-optimistic to expect either necessary and

sufficient conditions for causal statements in counterfactual terms or necessary and

sufficient conditions for counterfactual statements in causal terms. Even so,

counterfactuals surely play a crucial role in our causal thinking (see Harris 2000: 118-139

and Byrne 2005: 100-128 for some empirical discussion). Only extreme sceptics deny the

cognitive value of causal thought.

At a more theoretical level, claims of nomic necessity support counterfactual

conditionals. If it is a law that property P implies property Q, then typically if something

were to have P, it would have Q. If we can falsify the counterfactual in a specific case,

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perhaps by using better-established laws, we thereby falsify that claim of lawhood. We

sometimes have enough evidence to establish what the result of an experiment would be

without actually doing the experiment: that matters in a world of limited resources.

Counterfactual thought is deeply integrated into our empirical thought in general.

Although that consideration will not deter the most dogged sceptics about our knowledge

of counterfactuals, it indicates the difficulty of preventing such scepticism from

generalizing implausibly far, since our beliefs about counterfactuals are so well-

integrated into our general knowledge of our environment. I proceed on the assumption

that we have non-trivial knowledge of counterfactuals.

§2. In discussing the epistemology of counterfactuals, I assume no particular theory of

their compositional semantics, although I sometimes use the Stalnaker-Lewis approach

for purposes of illustration and vividness. That evasion of semantic theory might seem

dubious, since it is the semantics which determines what has to be known. However, we

can go some way on the basis of our pretheoretical understanding of such conditionals in

our native language. Moreover, the best developed formal semantic theories of

counterfactuals use an apparatus of possible worlds or situations at best distantly related

to our actual cognitive processing. While that does not refute such theories, which

concern the truth-conditions of counterfactuals, not how subjects attempt to find out

whether those truth-conditions obtain, it shows how indirect the relation between the

semantics and the epistemology may be. When we come to fine-tune our epistemology of

counterfactuals, we may need an articulated semantic theory, but at a first pass we can

make do with some sketchy remarks about their epistemology while remaining neutral

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over their deep semantic analysis. As for the psychological study of the processes

underlying our assessment of counterfactual conditionals, it remains in a surprisingly

undeveloped state, as recent authors have complained (Evans and Over 2004: 113-131).

Start with an example. You are in the mountains. As the sun melts the ice, rocks

embedded in it are loosened and crash down the slope. You notice one rock slide into a

bush. You wonder where it would have ended if the bush had not been there. A natural

way to answer the question is by visualizing the rock sliding without the bush there, then

bouncing down the slope. You thereby come to know this counterfactual:

(6) If the bush had not been there, the rock would have ended in the lake.

You could test that judgment by physically removing the bush and experimenting with

similar rocks, but you know (6) even without performing such experiments. Semantically,

the counterfactual about the past is independent of claims about future experiments (for a

start, the slope is undergoing continual small changes).

Somehow, you came to know the counterfactual by using your imagination. That

sounds puzzling if one conceives the imagination as unconstrained. You can imagine the

rock rising vertically into the air, or looping the loop, or sticking like a limpet to the

slope. What constrains imagining it one way rather than another?

You do not imagine it those other ways because your imaginative exercise is

radically informed and disciplined by your perception of the rock and the slope and your

sense of how nature works. The default for the imagination may be to proceed as

‘realistically’ as it can, subject to whatever deviations the thinker imposes by brute force:

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here, the absence of the bush. Thus the imagination can in principle exploit all our

background knowledge in evaluating counterfactuals. Of course, how to separate

background knowledge from what must be imagined away in imagining the antecedent is

Goodman’s old, deep problem of cotenability (1955). For example, why don’t we bring

to bear our background knowledge that the rock did not go far, and imagine another

obstacle to its fall? Difficult though the problem is, it should not make us lose sight of our

considerable knowledge of counterfactuals: our procedures for evaluating them cannot be

too wildly misleading.

Can the imaginative exercise be regimented as a piece of reasoning? We can

undoubtedly assess some counterfactuals by straightforward reasoning. For instance:

(7) If twelve people had come to the party, more than eleven people would

have come to the party.

We can deduce the consequent ‘More than eleven people came to the party’ from the

antecedent ‘Twelve people came to the party’, and assert (7) on that basis. Similarly, it

may be suggested, we can assert (6) on the basis of inferring its consequent ‘The rock

ended in the lake’ from the premise ‘The bush was not there’, given auxiliary premises

about the rock, the mountainside and the laws of nature.

At the level of formal logic, we have the corresponding plausible and widely

accepted closure principle that, given a derivation of C from BB1, …, BnB , we can derive the

counterfactual conditional A □→ C from the counterfactual conditionals A □→ B1, ….,

A □→ Bn; in other words, the counterfactual consequences of a supposition A are closed

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under logical consequence (Lewis calls this ‘Deduction within Conditionals’, 1986: 132).

With the uncontroversial reflexivity principle A □→ A, it follows that, given a derivation

of C from A alone, we can derive A □→ C from the null set of premises.

We cannot automatically extend the closure rule to the case of auxiliary premises,

for since we can derive an arbitrary conclusion C from an arbitrary premise A with C as

auxiliary premise, we could then derive A □→ C from the auxiliary premise C alone: but

that is in effect the invalid principle that any truth is a counterfactual consequence of any

supposition whatsoever. Auxiliary premises cannot always be copied into the scope of

counterfactual suppositions (the problem of cotenability again).

Even with this caution, the treatment of the process by which we reach

counterfactual judgments as inferential is problematic in several ways.

First, a technical problem: not every inference licenses us to assert the

corresponding counterfactual, even when the inference is deductive and the auxiliary

premises are selected appropriately. For the consequent of (1) is a logical truth (count it

vacuously true if Jones shows no symptoms):

(8) Jones shows just exactly those symptoms which he does in fact show.

Thus (8) follows from any premises, including (2), the antecedent of (1); but we cannot

assert (1) on the basis of that trivial deduction alone, independently of which symptoms

Jones does in fact show. This is related to Kaplan’s point that the rule of necessitation

fails in languages with terms such as ‘actually’ (1989). The logical truth of (8) does not

guarantee the logical truth, or even truth, of (9):

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(9) It is necessary that Jones shows just exactly those symptoms which he does in fact

show.

For it is contingent that Jones shows just exactly those symptoms which he does in fact

show.2 But let us assume that this technical problem can be solved by a restriction on the

type of reasoning from antecedent to consequent that can license a counterfactual, and on

the closure principle above, like the restriction on the type of reasoning that licenses the

necessitation of its conclusion.

A more serious problem is that the putative reasoner may lack general-purpose

cognitive access to the auxiliary premises of the putative reasoning. In particular, the

required folk physics may be stored in the form of some analogue mechanism, perhaps

embodied in a connectionist network, which the subject cannot articulate in propositional

form. Normally, a subject who uses negation and derives a conclusion from some

premises can at least entertain the negation of a given premise, whether or not they are

willing to assert it, perhaps on the basis of the other premises and the negation of the

conclusion. Our reliance on folk physics does not enable us to entertain its negation. This

strains the analogy with explicit reasoning.

The third problem is epistemological. Normally, someone who believes a

conclusion on the sole basis of deduction from some premises knows the conclusion only

if they know the premises. As a universally generalized theory, folk physics is

presumably strictly speaking false: its predictions are inaccurate in some circumstances.

Consequently, it is not known. But the conclusion that no belief formed on the basis of

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folk physics constitutes knowledge is wildly sceptical. For folk physics is reliable enough

in many circumstances to be used in the acquisition of knowledge, for example that the

cricket ball will land in that field. Thus we should not conceive folk physics as a premise

of that conclusion. Nor should we conceive some local fragment of folk physics as the

premise. For it would be quite unmotivated to take an inferential approach overall while

refusing to treat this local fragment as itself derived from the general theory of folk

physics. We should conceive folk physics as a locally but not globally reliable method of

belief formation, not as a premise.

The preceding reasons motivate the attempt to understand the imaginative

exercises by which we judge counterfactuals like (6) as not purely inferential. An

attractive suggestion is that some kind of simulation is involved: the difficulty is to

explain what that means. It is just a hint of an answer to say that in simulation cognitive

faculties are run off-line. The cognitive faculties that would be run on-line to evaluate A

and B as free-standing sentences are run off-line in the evaluation of the counterfactual

conditional A □→ B.3 This suggests that the cognition has a roughly compositional

structure. Our capacity to handle A □→ B embeds our capacities to handle A and B, and

our capacity to handle the counterfactual conditional operator involves a general capacity

to go from capacities to handle the antecedent and the consequent to a capacity to handle

the whole conditional. Here the capacity to handle an expression generally comprises

more than mere linguistic understanding of it, since it involves ways of assessing its

application that are not built into its meaning. But it virtually never involves a decision

procedure that enables us always to determine the truth-values of every sentence in which

the expression principally occurs, since we lack such decision procedures. Of course, we

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can sometimes take shortcuts in evaluating counterfactual conditionals. For instance, we

can know that A □→ A is true even if we have no idea how to determine whether A is

true. Nevertheless, the compositional structure just described seems more typical.

How do we advance from capacities to handle the antecedent and the consequent

to a capacity to handle the whole conditional? ‘Off-line’ suggests that the most direct

links with perception have been cut, but that vague negative point does not take us far.

Perceptual input is crucial to the evaluation of counterfactuals such as (1) and (6).

The best developed simulation theories concern our ability to simulate the mental

processes of other agents (or ourselves in other circumstances), putting ourselves in their

shoes, as if thinking and deciding on the basis of their beliefs and desires (see for

example Davies and Stone 1995, Nichols and Stich 2003). Such cognitive processes may

well be relevant to the evaluation of counterfactuals about agents. Moreover, they would

involve just the sort of constrained use of the imagination indicated above. How would

Mary react if you asked to borrow her car? You could imagine her immediately shooting

you, or making you her heir; you could even imagine reacting like that from her point of

view, by imagining having sufficiently bizarre beliefs and desires. But you do not. Doing

so would not help you determine how she really would react. Presumably, what you do is

to hold fixed her actual beliefs and desires (as you take them to be just before the

request); you can then imagine the request from her point of view, and think through the

scenario from there. Just as with the falling rock, the imaginative exercise is richly

informed and disciplined by your sense of what she is like.

How could mental simulation help us evaluate a counterfactual such as (6), which

does not concern an agent? Even if you somehow put yourself in the rock’s shoes,

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imagining first-personally being that shape, size and hardness and bouncing down that

slope, you would not be simulating the rock’s reasoning and decision-making. Thinking

of the rock as an agent is no help in determining its counterfactual trajectory. A more

natural way to answer the question is by imagining third-personally the rock falling as it

would visually appear from your actual present spatial position; you thereby avoid the

complex process of adjusting your current visual perspective to the viewpoint of the rock.

Is that to simulate the mental states of an observer watching the rock fall from your

present position?4 By itself, that suggestion explains little. For how do we know what to

simulate the observer seeing next? But that question is not unanswerable. For we have

various propensities to form expectations about what happens next: for example, to

project the trajectories of nearby moving bodies into the immediate future (otherwise we

could not catch balls). Perhaps we simulate the initial movement of the rock in the

absence of the bush, form an expectation as to where it goes next, feed the expected

movement back into the simulation as seen by the observer, form a further expectation as

to its subsequent movement, feed that back into the simulation, and so on. If our

expectations in such matters are approximately correct in a range of ordinary cases, such

a process is cognitively worthwhile. The very natural laws and causal tendencies which

our expectations roughly track also help to determine which counterfactual conditionals

really hold.

However, talk of simulating the mental states of an observer may suggest that the

presence of the observer is part of the content of the simulation. That does not fit our

evaluation of counterfactuals. Consider:

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(10) If there had been a tree on this spot a million years ago, nobody would have

known.

Even if we visually imagine a tree on this spot a million years ago, we do not

automatically reject (10) because we envisage an observer of the tree. We may imagine

the tree as having a certain visual appearance from a certain viewpoint, but that is not to

say that we imagine it as appearing to someone at that viewpoint. For example, if we

imagine the sun as shining from behind that viewpoint, by imagining the tree’s shadow

stretching back from the tree, we are not obliged to imagine either the observer’s shadow

stretching towards the tree or the observer as perfectly transparent.5 Nor, when we

consider (10), are we asking whether if we had believed that there was a tree on this spot

a million years ago, we would have believed that nobody knew.6 It may be better not to

think of the simulation as specifically mental simulation at all.

Of course, for many counterfactuals the relevant expectations are not hardwired

into us in the way that those concerning the trajectories of fast-moving objects around us

may need to be. Our knowledge that if a British general election had been called in 1948

the Communists would not have won may depend on an off-line use of our capacity to

predict political events. Still, where our more sophisticated capacities to predict the future

are reliable, so should be corresponding counterfactual judgments. In these cases too,

simulating the mental states of an imaginary observer seems unnecessary.

The off-line use of expectation-forming capacities to judge counterfactuals

corresponds to the widespread picture of the semantic evaluation of those conditionals as

‘rolling back’ history to shortly before the time of the antecedent, modifying its course by

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stipulating the truth of the antecedent and then rolling history forward again according to

patterns of development as close as possible to the normal ones to test the truth of the

consequent (compare Lewis 1979). Not all counterfactual conditionals can be so

evaluated, since the antecedent need not concern a limited time: in evaluating the claim

that space-time has ten dimensions, a scientist can sensibly ask whether if it were true the

actually observed phenomena would have occurred. Explicit reasoning may play a much

larger role in the evaluation of such conditionals.

Reasoning and prediction do not exhaust our capacity to evaluate counterfactuals.

If twelve people had come to the party, would it have been a large party? To answer, one

does not imagine a party of twelve people and then predict what would happen next. The

question is whether twelve people would have constituted a large party, not whether they

would have caused one. Nor is the process of answering best conceived as purely

inferential, if one has no special antecedent beliefs as to how many people constitute a

large party, any more than the judgment whether the party is large is purely inferential

when made at the party. Rather, in both cases one must make a new judgment, even

though it is informed by what one already believes or imagines about the party. To call

the new judgment ‘inferential’ simply because it is not made independently of all the

thinker’s prior beliefs or suppositions is to stretch the term ‘inferential’ beyond its useful

span. At any rate, the judgment cannot be derived from the prior beliefs or suppositions

purely by the application of general rules of inference. For example, even if you have the

prior belief that a party is large if and only if it is larger than the average size of a party,

in order to apply it to the case at hand you also need to have a belief as to what the

average size of a party is; if you have no prior belief as to that, and must form one by

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inference, an implausible regress threatens, for you do not have the statistics of parties in

your head. Similarly, if you try to judge whether this party is large by projecting

inductively from previous judgments as to whether parties were large, that only pushes

the question back to how those previous judgments were made.

In general, our capacity to evaluate counterfactuals recruits all our cognitive

capacities to evaluate sentences. A quick proof of this uses the assumption that a

counterfactual with a true antecedent has the same truth-value as its consequent, for then

any sentence A is logically equivalent to T □→ A, where T is a trivial tautology; so any

serious cognitive work needed to evaluate A is also needed to evaluate T □→ A.7

We can schematize the process of evaluating a counterfactual conditional thus:

the thinker imaginatively supposes the antecedent and counterfactually develops the

supposition, adding further judgments within the supposition by reasoning, off-line

predictive mechanisms and other off-line judgments. To a first approximation: if the

development eventually leads us to add the consequent, we assent to the conditional; if

not, we dissent from it. Of course, this initial sketch is much too crude, in several ways.

We may not be confident enough about the background conditions to decide for or

against the conditional. Even if we are confident enough in that respect, if the consequent

has not emerged after a given period of development the question remains whether it will

emerge in the course of further development, for lines of reasoning can be continued

indefinitely from any given premise. To reach a negative conclusion, we must in effect

judge that if the consequent were ever going to emerge it would have done so by now (for

example, we may have been smoothly fleshing out a scenario incompatible with the

consequent with no hint of difficulty). A further over-simplification was that we develop

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the initial supposition only once: if we find various different ways of imagining the

antecedent holding equally good, we may try developing several of them, to see whether

they all yield the consequent. For example, if in considering (10) you initially imagine a

palm tree, you do not immediately judge that if there had been a tree on this spot a

million years ago it would have been a palm tree, because you know that you can equally

easily imagine a fir tree. Although far more needs to be said, these remarks may at least

start us in the right direction.

Despite its discipline, our imaginative evaluation of counterfactual conditionals is

manifestly fallible. We can easily misjudge their truth-values, through background

ignorance or error, and distortions of judgment. But such fallibility is the common lot of

human cognition. Our use of the imagination in evaluating counterfactuals is practically

indispensable. Rather than cave in to scepticism, we should admit that our methods

sometimes yield knowledge of counterfactuals.

§3. How does the epistemology of counterfactual conditionals bear on the epistemology

of metaphysical modality? We can approach this question by formulating two plausible

constraints on the relation between counterfactual conditionals and metaphysical

modalities. Henceforth, ‘necessary’ and ‘possible’ will be used for the metaphysical

modalities unless otherwise stated.

First, the strict conditional implies the counterfactual conditional:

NECESSITY □(A B) ⊃ ⊃ (A □→ B)

21

Suppose that A could not have held without B holding too; then if A had held, B would

also have held. In terms of possible worlds semantics for these operators along the lines

of Lewis (1973) or Stalnaker (1968): if all A worlds are B worlds, then any closest A

worlds are B worlds. More precisely, if all A worlds are B worlds, then either there are no

A worlds or there is an A world such that any A world at least as close as it is to the

actual world is a B world.

Second, the counterfactual conditional transmits possibility:

POSSIBILITY (A □→ B) ⊃ (◊A ⊃ ◊B)

Suppose that if A had held, B would also have held; then if it is possible for A to hold, it

is also possible for B to hold. In terms of worlds: if any closest A worlds are B worlds,

and there are A worlds, then there are also B worlds. More precisely, if either there are no

A worlds or there is an A world such that any A world at least as close as it is to the

actual world is a B world, then if there is an A world there is also a B world.

Together, NECESSITY and POSSIBILITY sandwich the counterfactual

conditional between two modal conditions. But they do not squeeze it very tight, for

◊A ◊B is much weaker than □(A ⊃ ⊃ B): although the latter entails the former in any

normal modal logic, the former is true and the latter false whenever B is possible without

being a necessary consequence of A, for example when A and B are modally

independent.

Although NECESSITY and POSSIBILITY determine no necessary and sufficient

condition for the counterfactual conditional in terms of necessity and possibility, they

22

yield necessary and sufficient conditions for necessity and possibility in terms of the

counterfactual conditional.

We argue thus. Let be a contradiction. As a special case of NECESSITY: ⊥

(11) □(¬A ) (¬A □→ ⊃ ⊥ ⊃ ⊥ )

By elementary (normal) modal logic, since a truth-functional consequence of something

necessary is itself necessary:

(12) □A □(¬A ) ⊃ ⊃ ⊥

From (11) and (12) by transitivity of the material conditional:

(13) □A (¬A □→ ) ⊃ ⊥

Similarly, as a special case of POSSIBILITY:

(14) (¬A □→ ) (◊¬A ⊃ ◊⊥ ⊃ ⊥ )

By elementary (normal) modal logic, since the possibility of a contradiction is itself

inconsistent, and necessity is the dual of possibility (being necessary is equivalent to

having an impossible negation):

23

(15) (◊¬A ◊ ) □A ⊃ ⊥ ⊃

From (14) and (15) by transitivity:

(16) (¬A □→ ) □A ⊥ ⊃

Putting (13) and (16) together:

(17) □A (¬A □→ ) ≡ ⊥

The necessary is that whose negation counterfactually implies a contradiction. Since

possibility is the dual of necessity (being possible is equivalent to having an unnecessary

negation), (17) yields a corresponding necessary and sufficient condition for possibility,

once a double negation in the antecedent of the counterfactual has been eliminated.

(18) ◊A ¬(A □→ ) ≡ ⊥

The impossible is that which counterfactually implies a contradiction; the possible is that

which does not. In (17) and (18), the difference between necessity and possibility lies

simply in the scope of negation.

Without assuming a specific framework for the semantics of counterfactuals (in

particular, that of possible worlds), we can give a simple semantic rationale for (17) and

(18), based on the idea of vacuous truth. That some true counterfactuals have impossible

24

antecedents is clear, for otherwise A □→ A would fail when A was impossible. Make

two generally accepted assumptions about the distinction between vacuous and non-

vacuous truth: (a) B □→ C is vacuously true if and only if B is impossible (this could be

regarded as a definition of ‘vacuously’ for counterfactuals); (b) B □→ C is non-

vacuously true only if C is possible. The truth of (17) and (18) follows, given normal

modal reasoning. If □A is true, then ¬A is impossible, so by (a) ¬A □→ is vacuously

true; conversely, if ¬A □→ ⊥ is true, then by (b) it is vacuously true, so by (a) ¬A is

impossible, so □A is true. Similarly, if ◊A is true, then A is not impossible, so by (a)

⊥

A □→ ¬ is not vacuously true, and by (b) not non-vacuously true, so ¬(A □→ ) is

true; if ◊A is not true, then A is impossible, so by (a) A □→ ¬

⊥ ⊥

⊥ is vacuously true, so

¬(A □→ ) is not true. ⊥

Given that the equivalences (17) and (18) are logically true, metaphysically modal

thinking is logically equivalent to a special case of counterfactual thinking, and the

epistemology of the former is tantamount to a special case of the epistemology of the

latter. Whoever has what it takes to understand the counterfactual conditional and the

elementary logical auxiliaries ¬ and ⊥ has what it takes to understand possibility and

necessity operators.

The definability of necessity and possibility in terms of counterfactual

conditionals was recognized long ago. It is easy to show from the closure and reflexivity

principles for □→ in §2 that A □→ ⊥ is logically equivalent to A □→ ¬A. Thus (17) and

(18) generate two new equivalences:

(19) □A (¬A □→ A) ≡

25

(20) ◊A ¬(A □→ ¬A) ≡

The necessary is that which is counterfactually implied by its own negation; the possible

is that which does not counterfactually imply its own negation. Stalnaker (1968) used

(19) and (20) to define necessity and possibility, although his reading of the conditional

(with a different notation) was not exclusively counterfactual. Lewis (1973a: 25) used

(17) and (18) themselves to define necessity and possibility in terms of the counterfactual

conditional. However, such definitions seem to have been treated as convenient

notational economies, their potential philosophical significance unnoticed (Hill 2006 is a

recent exception).

If we permit ourselves to quantify into sentence position (‘propositional

quantification’), we can formulate another pair of variants on (17) and (18) that may

improve our feel for what is going on.8 On elementary assumptions about the logic of

such quantifiers and of the counterfactual conditional, ¬A □→ A is provably equivalent

to ∀ p (p □→ A): something is counterfactually implied by its negation if and only if it is

counterfactually implied by everything. Thus (19) and (20) generate these equivalences

too:

(21) □A ∀ p (p □→ A) ≡

(22) ◊A p ¬(p □→ ¬A) ≡ ∃

26

According to (21), something is necessary if and only if whatever were the case, it would

still be the case (see also Lewis 1986: 23). That is a natural way of explaining informally

what metaphysically necessity is. According to (22), something is possible if and only if

it is not such that it would fail in every eventuality.

Since the right-hand sides of (17), (19) and (21) are not strictly synonymous with

each other, given the differences in their semantic structure, they are not all strictly

synonymous with □A. Similarly, since the right-hand sides of (18), (20) and (22) are not

strictly synonymous with each other, they are not all strictly synonymous with ◊A.

Indeed, we have no sufficient reason to regard any of the equivalences as strict

synonymies. That detracts little from their philosophical significance, for failure of strict

synonymy does not imply failure of logical equivalence. The main philosophical

concerns about possibility and necessity apply equally to anything logically equivalent to

possibility or necessity. A non-modal analogy: ¬A is logically equivalent to A , but

presumably they are not strictly synonymous; nevertheless, once we have established that

a creature can handle and , we have established that it can handle something

logically equivalent to negation, which answers the most interesting questions about its

ability to handle negation. We should find the mutual equivalence of (17), (19) and (21),

and of (18), (20) and (22) reassuring, for it shows the robustness of the modal notions

definable from the counterfactual conditional, somewhat as the equivalence of the various

proposed definitions of ‘computable function’ showed the robustness of that notion.

⊃ ⊥

⊃ ⊥

If we treat (17) and (18) like definitions of □ and ◊ for logical purposes, and

assume some elementary principles of the logic of counterfactuals, then we can establish

the main principles of elementary modal logic for □ and ◊. For example, we can show

27

that what follows from necessary premises is itself necessary. Given that counterfactual

conditionals obey modus ponens (or even weaker assumptions), we can show that what is

necessary is the case. We can also check that the principles NECESSITY and

POSSIBILITY, which we used to establish (17) and (18), do indeed hold under the latter

characterizations of necessity and possibility. Under much stronger assumptions about

the logic of the counterfactual conditional, we can also establish much stronger principles

of modal logic, such as the S5 principle that what is possible is necessarily possible. Such

connections extend to quantified modal logic. The logic of counterfactual conditionals

smoothly generates the logic of the modal operators. Technical details are omitted here.

In particular, the proposed conception of modality makes quantification into the

scope of modal operators tantamount to a special case of quantification into

counterfactual contexts, as in (23) and (24):

(23) Everyone who would have benefited if the measure had passed voted for it.

(24) Where would the rock have landed if the bush had not been there?

Thus challenges to the intelligibility of claims of de re necessity are tantamount to

challenges to the intelligibility of counterfactuals such as (23) and (24). But (23) and (24)

are evidently intelligible.

Given (17) and (18), we should expect the epistemology of metaphysical modality

to be a special case of the epistemology of counterfactuals. Far from being sui generis,

the capacity to handle metaphysical modality is an ‘accidental’ byproduct of the cognitive

28

mechanisms which provide our capacity to handle counterfactual conditionals. Since our

capacity for modal thinking cannot be isolated from our capacity for ordinary thinking

about the natural world, which involves counterfactual thinking, sceptics cannot excise

metaphysical modality from our conceptual scheme without loss to ordinary thought

about the natural world, for the former is implicit in the latter.

A useful comparison is with the relation between logical consequence and logical

truth. Consider some agents who reason in simple ways about themselves and their

environment, perhaps using rules of inference formalizable in a Gentzen-style natural

deduction calculus, perhaps in some less sophisticated way. The practical value of their

reasoning skill is that they can move from ordinary empirical premises to ordinary

empirical conclusions in ways that always preserve truth, thereby extending their

knowledge of mundane matters (see Schechter 2006 for relevant discussion). In doing so,

they need never use logically true sentences. Nevertheless, the cognitive capacity that

enables them to make these transitions between empirical sentences also enables them, as

a special case, an ‘accidental’ byproduct, to deduce logical truths from the null set of

premises. Highly artificial moves would be needed to block these bonus deductions; such

ad hoc restrictions would come at the price of extra computational complexity for no

practical gain. Likewise at the semantic level: the simplest compositional semantics that

enables us to negate and conjoin empirical sentences also enables us to formulate logical

truths and falsehoods, even if we have hitherto lacked any interest in doing so. By good

fortune, everything is already in place for the logician to evaluate logical truths and

falsehoods (at least in first-order logic, since it is complete). The philosopher’s position

with respect to metaphysical modality is not very different.

29

Discussions of the epistemology of modality often focus on imaginability or

conceivability as a test of possibility while ignoring the role of the imagination in the

assessment of mundane counterfactuals. In doing so, they omit the appropriate context for

understanding the relation between modality and the imagination. For instance, scorn is

easily poured on imagination as a test of possibility: it is imaginable but not possible that

water does not contain oxygen, except in artificial senses of ‘imaginable’ that come apart

from possibility in other ways, and so on. Imagination can be made to look cognitively

worthless. Once we recall its fallible but vital role in evaluating counterfactual

conditionals, we should be more open to the idea that it plays such a role in evaluating

claims of possibility and necessity. At the very least, we cannot expect an adequate

account of the role of imagination in the epistemology of modality if we lack an adequate

account of its role in the epistemology of counterfactuals.

On the simplest version of the account in §2, we accept A □→ B when our

counterfactual development of the supposition A generates B; we reject A □→ B when

our counterfactual development of A fails to generate B (in a reasonable time). Thus, by

(17), we accept □A when our counterfactual development of the supposition ¬A

generates a contradiction; we reject □A when our counterfactual development of ¬A fails

to generate a contradiction (in a reasonable time). Similarly, by (18), we accept ◊A when

our counterfactual development of the supposition A fails to generate a contradiction (in

a reasonable time); we reject ◊A when our counterfactual development of A generates a

contradiction. Thus our fallible imaginative evaluation of counterfactuals has a

conceivability test for possibility and an inconceivability test for impossibility as fallible

special cases. Such conceivability and inconceivability will be subject to the same

30

constraints, whatever they are, as counterfactual conditionals in general, concerning

which parts of our background information are held fixed. If we know enough chemistry,

our counterfactual development of the supposition that gold is the element with atomic

number 79 will generate a contradiction. The reason is not simply that we know that gold

is the element with atomic number 79, for we can and must vary some items of our

knowledge under counterfactual suppositions. Rather, general constraints on the

development of counterfactual suppositions require us to hold such constitutive facts

fixed.

A nuanced account of our handling of counterfactuals is likely to predict that we

are more reliable in evaluating some kinds than others. For example, we may well be

more reliable in evaluating counterfactuals whose antecedents involve small departures

from the actual world than in evaluating those whose antecedents involve much larger

departures. We may be correspondingly more reliable in evaluating the possibility of

everyday scenarios than of ‘far-out’ ones, and extra caution may be called for in the latter

case. At the limit, actuality is often the best argument for possibility. But current

philosophical practice already shows some sensitivity to such considerations. We may be

more confident of the possibility of more or less realistic thought experiments in

epistemology and moral philosophy than of more radically strange ones in metaphysics.

More explicit consideration of the link between modal thought and counterfactual

thought may lead to further refinements of our practice. But the use of imagination to

evaluate philosophical claims of possibility and necessity is not illegitimate in principle,

any more than is its use to evaluate mundane counterfactuals.

31

What does the envisaged assimilation of modality to counterfactual conditionals

imply for the status of modal judgments as knowable a priori or only a posteriori? Some

counterfactual conditions look like paradigms of a priori knowability: for example (7),

whose consequent is a straightforward deductive consequence of its antecedent. Others

look like paradigms of what can be known only a posteriori: for example, that if I had

searched in my pocket five minutes ago I would have found a coin. But those are easy

cases.

Standard discussions of the a priori distinguish between two roles that experience

plays in cognition, one evidential, one enabling. Experience is held to play an evidential

role in my visual knowledge that this shirt is green, but a merely enabling role in my

knowledge that all green things are coloured: I needed it only to acquire the concepts

green and coloured, without which I could not even raise the question whether all green

things are coloured. Knowing a priori is supposed to be incompatible with an evidential

role for experience, so my knowledge that this shirt is green is not a priori; but

compatible with an enabling role for experience, so my knowledge that all green things

are coloured can still be a priori. However, in our imagination-based knowledge of

counterfactuals, experience can play a role that is neither strictly evidential nor purely

enabling. For it can mould the ways in which we later imagine and judge, beyond what is

needed to grasp the relevant concepts, without surviving as part of our total evidence.

Here is an example. I acquire the words ‘inch’ and ‘centimetre’ independently of

each other. Through experience, I learn to make naked eye judgments of distances in

inches or centimetres with moderate reliability. When things go well, such judgments

amount to knowledge: a posteriori knowledge, of course. For example, I know a

32

posteriori that two marks in front of me are at most two inches apart. Now I deploy the

same faculty off-line to make a counterfactual judgment:

(25) If these marks had been at least nine inches apart, they would have been at

least nineteen centimetres apart.

In judging (25), I do not use a conversion ratio between inches and centimetres to make a

calculation. In the example I know no such ratio. Rather, I visually imagine the two

marks nine inches apart, and use my ability to judge distances in centimetres visually off-

line to judge under the counterfactual supposition that the marks are at least nineteen

centimetres apart. With this large margin for error, my judgment is reliable. Thus I know

(25). Do I know it a priori or a posteriori? Experience plays no direct evidential role in

my judgment. I do not consciously or unconsciously recall memories of distances

encountered in perception, nor do I deduce (25) from general principles that I have

inductively or abductively gathered from experience: §2 noted obstacles to assimilating

counterfactual thinking to reasoning. Nevertheless, the causal role of past experience in

my judgment of (25) far exceeds enabling me to grasp the concepts in (25). Someone

could easily have enough experience to understand (25) without being reliable enough in

their judgments of distance to know (25).

If we classify my knowledge of (25) in the envisaged circumstances as a priori,

because experience plays no strictly evidential role, the danger is that far too much will

count as a priori. Experience can mould my judgment in many ways without playing a

direct evidential role. But if we classify my knowledge of (25) as a posteriori, because

33

experience plays more than a purely enabling role, that may apply to many

philosophically significant modal judgments too. Of course, Kripke has argued strongly

for a category of necessary truths knowable only a posteriori, such as ‘Gold is the

element with atomic number 79’; ‘It is necessary that gold is the element with atomic

number 79’ would then be knowable only a posteriori too. The present suggestion is

intended far more widely than that. For example:

(26) It is necessary that whoever knows something believes it.

(27) If Mary knew that it was raining, she would believe that it was raining.

Knowledge of truths such as (26) and (27) is usually regarded as a priori, even by those

who accept the category of the necessary a posteriori. The experiences through which we

learned to distinguish in practice between belief and non-belief and between knowledge

and ignorance play no strictly evidential role in our knowledge of (26) and (27).

Nevertheless, their role may be more than purely enabling. Many philosophers, native

speakers of English, have denied (26) (Shope 1983: 171-192 has a critical survey). They

are not usually or plausibly accused of failing to understand the words ‘know’ and

‘believe’. Why should not subtle differences between two courses of experience, each of

which sufficed for coming to understand ‘know’ and ‘believe’, make for differences in

how test cases are imagined, just large enough to tip honest judgments in opposite

directions? Whether knowledge of (26) and (27) is available to one may thus be highly

sensitive to personal circumstances.

34

If that picture is on the right lines, should we conclude that modal knowledge is a

posteriori? Not if that suggests that (26) and (27) are inductive or abductive conclusions

from perceptual data. In such cases, the question ‘A priori or a posteriori?’ is too crude to

be of much epistemological use. The point is not that we cannot draw a line somewhere

with traditional paradigms of the a priori on one side and traditional paradigms of the a

posteriori on the other. Surely we can; the point is that doing so yields little insight. The

distinction is handy enough for a rough initial description of epistemic phenomena; it is

out of place in a deeper theoretical analysis, because it obscures more significant

epistemic patterns.9

§4. It is time to consider objections to the preceding account.

Objection: Knowledge of counterfactuals cannot explain modal knowledge,

because the former depends on the latter. More specifically, in developing a

counterfactual supposition, we make free use of what we take to be necessary truths, but

not of what we take to be contingent truths. Thus we rely on a prior stock of modal

knowledge or belief. The principle NECESSITY above illustrates how we do this.

Reply: Once we take something to be a necessary truth, of course we can use it in

developing further counterfactual suppositions. But that does nothing to show that we

have any special cognitive capacity to handle modality independent of our general

cognitive capacity to handle counterfactual conditionals. If we start only with the latter,

just as envisaged above, it will generate knowledge of various modal truths, which can in

turn be used to develop further counterfactual suppositions, in a recursive process. For

example, we need not judge that it is metaphysically necessary that gold is the element

35

with atomic number 79 before invoking the proposition that gold is the element with

atomic number 79 in the development of a counterfactual supposition. Rather, projecting

constitutive matters such as atomic numbers into counterfactual suppositions is part of

our general way of assessing counterfactuals. The judgment of metaphysical necessity

originates as the output of a procedure of that kind; it is not an independently generated

input.

Objection: The account associates metaphysical modality with counterfactual

conditionals of a very peculiar kind: in the case of (17) and (18), those with an explicit

contradiction as their consequent. Why should a capacity to handle ordinary

counterfactuals confer a capacity to handle such peculiar ones too?

Reply: That is like asking why a capacity to handle inferences between complex

empirical sentences should confer a capacity to handle inferences involving logical truths

and falsehoods too. There is no easy way to have the former without the latter. More

specifically, developing a counterfactual supposition includes reasoning from it, and we

cannot always tell in advance when such reasoning will yield a contradiction (there are

surprises in logic). The undecidability of logical truth for first-order logic implies that

there is no total mechanical test for the consistency of first-order sentences. Thus the

inconsistent ones cannot be sieved out in advance (consider ‘In the next village there is a

barber who shaves all and only those in that village who do not shave themselves’).

Consequently, a general capacity to develop counterfactual suppositions must confer in

particular the capacity to develop those which subsequently turn out inconsistent.

Although the capacity may not be of uniform reliability, as already noted, the variation is

primarily with the antecedent of the counterfactual (the supposition under development),

36

not with its consequent (which is what is exceptional in (17) and (18)). In deductive

inference, our reasoning to contradictions (as in proof by reduction ad absurdum) is not

strikingly more or less reliable than the rest of our deductive reasoning.

Objection: The assumption about vacuous truth on which the account relies is

wrong (Nolan 1997). For some counterpossibles (counterfactuals with metaphysically

impossible antecedents) are false, such as (28), uttered by someone who mistakenly

believes that he answered ‘13’ to ‘What is 5 + 7?’; in fact he answered ‘11’:

(28) If 5 + 7 were 13 I would have got that sum right.

Thus, contrary to (17), □A may be true while ¬A □→ ⊥ is false. In the argument for (17)

in §3, the objectionable premise is NECESSITY. If some worlds are metaphysically

impossible, and A is true at some of them but false at all metaphysically possible worlds,

while B is false at all worlds whatsoever, then every metaphysically possible A world is a

B world, but the closest A worlds are not B worlds.10 Similar objections apply to the

other purported equivalences (18)-(22).

Reply: If all counterpossibles were false, ◊A would be equivalent to A □→ A, for

the latter would still be true whenever A was possible; correspondingly, □A would be

equivalent to the dual ¬(¬A □→ ¬A) and one could carry out the programme of §3 using

the new equivalences. But that is presumably not what the objector has in mind. Rather,

the idea is that the truth-value of a counterpossible can depend on its consequent, so that

(28) is false while (29) is true:

37

(29) If 5 + 7 were 13 I would have got that sum wrong.

However, such examples are quite unpersuasive.

First, they tend to fall apart when thought through. For example, if 5 + 7 were 13

then 5 + 6 would be 12, and so (by another eleven steps) 0 would be 1, so if the number

of right answers I gave were 0, the number of right answers I gave would be 1.

Second, there are general reasons to doubt the supposed intuitions on which such

examples rely. We are used to working with possible antecedents, and given the

possibility of A, the incompatibility of B and C implies that A □→ B and A □→ C

cannot both be true. Thus by over-projecting from familiar cases we may take the

uncontentious (29) to be incompatible with (28). The logically unsophisticated make

analogous errors in quantificational reasoning. Given the evident truth of ‘Every golden

mountain is a mountain’, they think that ‘Every golden mountain is a valley’ is false,

neglecting the case of vacuous truth. Since the logic and semantics of counterfactual

conditionals is much less well understood, even the logically sophisticated may find

similar errors tempting. Such errors may be compounded by a tendency to confuse

negating a counterfactual conditional with negating its consequent, given the artificiality

of the constructions needed to negate the whole conditional unambiguously (‘it is not the

case that if …’). Thus the truth of A □→ ¬B (with A impossible) may be mistaken for the

truth of ¬(A □→ B) and therefore the falsity of A □→ B.

Some objectors try to bolster their case by giving examples of mathematicians

reasoning from an impossible supposition A (‘There are only finitely many prime

numbers’) in order to reduce it to absurdity. Such arguments can be formulated using a

38

counterfactual conditional, although they need not be. Certainly there will be points in the

argument at which it is legitimate to assert A □→ C (in particular, A □→ A) but

illegitimate to assert A □→ ¬C (in particular, A □→ ¬A). But of course that does not

show that A □→ ¬A is false. At any point in a mathematical argument there are infinitely

many truths that it is not legitimate to assert, because they have not yet been proved

(Lewis 1986: 24-6 pragmatically explains away some purported examples of false

counterfactuals with impossible antecedents).

We may also wonder what logic of counterfactuals the objectors envisage. If they

reject elementary principles of the pure logic of counterfactual conditionals, that is an

unattractive feature of their position. If they accept all those principles, then they are

committed to operators characterized as in (17) and (18) that exhibit all the logical

behaviour standardly expected of necessity and possibility. What is that modality, if not

metaphysical modality?

A final problem for the objection is this. Here is a paradigm of the kind of

counterpossible which the objector regards as false:

(30) If Hesperus had not been Phosphorus, Phosphorus would not have been

Phosphorus.

Since Hesperus is Phosphorus, it is metaphysically impossible that Hesperus is not

Phosphorus, by the necessity of identity. Nevertheless, the objectors are likely to insist

that in imaginatively developing the counterfactual supposition that Hesperus is not

Phosphorus, we are committed to the explicit denial of no logical truth, as in the

39

consequent of (30). According to them, if we do our best for the antecedent, we can

develop it into a logically coherent though metaphysically impossible scenario: it will

exclude ‘Phosphorus is not Phosphorus’. But they will presumably accept this trivial

instance of reflexivity:

(31) If Hesperus had not been Phosphorus, Hesperus would not have been Phosphorus.

In general, however, coreferential proper names are intersubstitutable in counterfactual

contexts. For example, the argument from (32) and (33) to (34) is unproblematically

valid:

(32) If the rocket had continued on that course, it would have hit Hesperus.

(33) Hesperus = Phosphorus.

(34) If the rocket had continued on that course, it would have hit Phosphorus.

Similarly, the argument from (31) and (33) to (30) should be valid. But (31) and (33) are

uncontentiously true. If the objector concedes that (30) is true after all, then there should

be an explanation of the felt resistance to it, compatible with its truth, and we may

reasonably expect that explanation to generalize to other purported examples of false

counterpossibles. On the other hand, if objectors reject (30), they must deny the validity

of the argument from (31) and (33) to (30). Thus they are committed to the claim that

40

counterfactual conditionals create opaque contexts for proper names (the same argument

could be given for other singular terms, such as demonstratives). But that is highly

implausible. (32) and (34) are materially equivalent because their antecedents and

consequents concern the same objects, properties and relations: it matters not that

different names are used, because the counterfactuals are not about such representational

features. But then exactly the same applies to (30) and (31). Their antecedents and

consequents too concern the same objects, properties and relations. That the antecedent of

(30) and (31) is in fact metaphysically impossible does not radically alter their subject

matter. The transparency of the counterfactual conditional construction concerns its

general logical form, not the specific content of the antecedent.

Under scrutiny, the case for false counterpossibles looks feeble.

Objection: Counterfactuals are desperately vague and context-sensitive;

equivalences such as (17) and (18) will infect □ and ◊, interpreted as metaphysical

modalities, with all that vagueness and context-sensitivity.

Reply: Infection is not automatic. For instance, within a Lewis-Stalnaker

framework, different readings or sharpenings of □→ may differ on the similarity ordering

of worlds while still agreeing on what worlds there are, so that the differences cancel out

in the right-hand sides of (17) and (18). Whether a given supposition counterfactually

implies a contradiction may be unclear to us; that does not imply that there is no right

answer.

Objection: It has been argued that counterfactual conditionals lack truth-values

(Edgington 2003, Bennett 2003: 252-6). If so, the assimilation of claims of metaphysical

possibility and necessity to counterfactuals will deprive such claims of truth-values.

41

Reply: The issues are too complex to discuss properly here, but the readily

intelligible occurrence of counterfactual conditionals embedded in the scope of other

operators as in (23) and (24) is hard to make sense of without attributing truth-values to

the embedded occurrences. Here is another example:

(35) Every field that would have been flooded if the dam had burst was ploughed.

(35) can itself be intelligibly embedded in more complex sentences in all the usual ways.

In order to understand how such embeddings work, we must assign truth-conditions to

(35); ad hoc treatments of a few particular embeddings are not enough. For (35) to have

truth-conditions, ‘field that would have been flooded if the dam had burst’ must have

application-conditions. Thus there must be a distinction between the fields to which

‘would have been flooded if the dam had burst’ applies and those to which it does not.

But that is just to say that there must be a distinction between the values of ‘x’ for which

‘If the dam had burst, x would have been flooded’ is true and those for which it is false.

That it is somewhat obscure what the truth-conditions of counterfactual conditionals are,

and that we sometimes make conflicting judgments about them, hardly shows that they

do not exist.

§5. The counterfactual conditional is of course not the only construction in ordinary use

that is closely related to metaphysical modality. Consider comments after a swiftly

extinguished fire in an explosives factory:

42

(36) There could have been a huge explosion.

(37) There could easily have been a huge explosion.

The truth-value of both (36) (so interpreted) and (37) depends on the location of the fire,

the precautions in place, and so on. The mere metaphysical possibility of a huge

explosion is insufficient to verify either (36) (so interpreted) or (37). The restricted nature

of the possibility is explicit in (37) with the word ‘easily’; it is implicit in the context of

(36).11 To discover the truth-value of (36) or (37), we need background information. We

may also need our imagination, in attempting to develop a feasible scenario in which

there is a huge explosion. We use the same general cognitive faculties as we do in

evaluating related counterfactual conditionals, such as (38):

(38) If the fire engine had arrived a minute later, there would have been a huge

explosion.

Judgments of limited possibility such as (36) (interpreted as above) and (37) have a

cognitive value for us similar to that of counterfactual conditionals such as (38).

Both (36) and (37) entail (39), although not vice versa:

(39) It is metaphysically possible that there was a huge explosion.

43

This is another way in which our ordinary cognitive capacities enable us to recognize that

something non-actual is nevertheless metaphysically possible. But we cannot reason from

the negation of (36) or of (37) to the negation of (39).

Can metaphysical possibility be understood as the limiting case of such more

restricted forms of possibility? Perhaps, but we would need some account of what

demarcates the relevant forms of possibility from irrelevant ones, such as epistemic

possibility. It also needs to be explained how, from the starting-point of ordinary thought,

we manage to single out the limiting case, metaphysical modality. The advantage of

counterfactual conditionals is that they allow us to single out the limiting case simply by

putting a contradiction in the consequent; contradictions can be formed in any language

with conjunction and negation Anyway, the connections with restricted possibility and

with counterfactual conditionals are not mutually exclusive, for they are not being

interpreted as rival semantic analyses, but rather as different cases in which the cognitive

mechanisms needed for one already provide for the other.

The epistemology of metaphysical modality requires no dedicated faculty of

intuition. It is simply a special case of the epistemology of counterfactual thinking, a kind

of thinking tightly integrated with our thinking about the spatio-temporal world. To deny

that such thinking ever yields knowledge is to fall into an extravagant scepticism. Here as

elsewhere, we can do philosophy on the basis of general cognitive capacities that are in

no deep way peculiarly philosophical.

44

Notes

* An earlier version of this paper was presented at the 2005 Erfurt conference on

philosophical knowledge. It developed out of Williamson 2005, especially 15-22, from

which it inherits numerous debts; Williamson 2004 explains other aspects of the

associated general account of philosophical knowledge. Further debts were acquired from

discussion after the presentation of that paper as a Presidential Address to the Aristotelian

Society and of related material in the Blackwell Brown lectures at Brown University, the

Anders Wedberg lectures at Stockholm University (where Sören Häggqvist and Anna-

Sara Malmgren commented on the relevant lectures), at the third meeting of the

Portugese Society for Analytic Philosophy in Lisbon and at conferences and workshops

at the Centre for Advanced Studies in the Norwegian Academy of Sciences in Oslo, the

Australian National University, Rutgers University and the universities of Bristol,

Munich and Rochester, at colloquia at the University of California at Los Angeles and the

universities of Arizona, Bologna, Heidelberg, Leeds, Nottingham, Turin and Warwick,

and in Oxford, although the ideas presented here were not always centre stage. Thanks to

all the individuals who have helped improve this paper, and in particular to Ann-Sara

Malmgren and Thomas Kroedel for many valuable discussions of its themes.

1 The large empirical literature on the affective role of counterfactuals and its

relation to learning from experience includes Kahneman and Tversky 1982, Roese and

Olson 1993, 1995 and Byrne 2005.

45

2 The phrase ‘does in fact show’ is read throughout as inside the scope of the

counterfactual conditional or modal operator, but as rigid, like ‘actually shows’. See

Williamson 2006 for relevant discussion.

3 Matters become more complicated if A or B itself contains a counterfactual

condition, as in ‘If she had murdered the man who would have inherited her money if she

had died, she would have been sentenced to life imprisonment if she had been convicted’.

4 See Goldman 1992: 24, discussed by Nichols, Stich, Leslie and Klein 1996: 53-

59.

5 The question is of course related to Berkeley’s claim that we cannot imagine an

unseen object. For discussion see Williams 1966, Peacocke 1985 and Currie 1995: 36-37.

6 A similar problem arises for what is sometimes called the Ramsey Test for

conditionals, on which one simulates belief in the antecedent and asks whether one then

believes the consequent. Goldman writes ‘When considering the truth value of “If X were

the case, then Y would obtain,” a reasoner feigns a belief in X and reasons about Y under

that pretense” (1992: 24). What Ramsey himself says is that when people ‘are fixing their

degrees of belief in q given p’ they ‘are adding p hypothetically to their stock of

knowledge and arguing on that basis about q’ (1978: 143), but he specifically warns that

‘the degree of belief in q given p’ does not mean the degree of belief ‘which the subject

46

would have in q if he knew p, or that which he ought to have’ (1978: 82; variables

interchanged). Of course, conditional probabilities bear more directly on indicative than

on subjunctive conditionals.

7 Lewis defends the assumption (1986: 26-31); Nozick rejects it to make the fourth

condition in his analysis of knowledge non-trivial (1981: 176). Bennett also rejects it

(2003: 239-40).

8 This quantification into sentence position need not be understood substitutionally.

In purely modal contexts it can be modeled as quantification over all sets of possible

worlds, even if not all of them are intensions of sentences that form the supposed

substitution class, although this modeling presumably fails for hyperintensional contexts

such as epistemic ones. A more faithful semantics for it might use non-substitutional

quantification into sentence position in the meta-language. Such subtleties are inessential

for present purposes.

9 This problem for the a priori/a posteriori distinction undermines arguments for

the incompatibility of semantic externalism with our privileged access to our own mental

states that appeal to the supposed absurdity of a priori knowledge of contingent features

of the external environment (McKinsey 1991).

10 Technically, NECESSITY fails on a semantics with similarity spheres for □→

that include some impossible worlds (inaccessible with respect to □). Conversely,

47

POSSIBILITY fails on a semantics with some possible worlds excluded from all

similarity spheres (see Lewis 1986: 16 on universality). Inaccessible worlds seem not to

threaten POSSIBILITY. For suppose that an A world w but no B world is accessible from

a world v. Then if A □→ B holds at v on the usual semantics, there is an A world x such

that every A world as close as x is to v is a B world. It follows that w is not as close as x is

to v and that x is inaccessible from v, which contradicts the plausible assumption that any

accessible world is at least as close as any inaccessible world.

11 On easy possibility see Sainsbury 1997, Peacocke 1999: 310-28 and Williamson

2000: 123-30. On the idea that natural language modals such as ‘can’ and ‘must’ advert

to contextually restricted ranges of possibilities see Kratzer 1977.

48

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